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• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (692 words) - 22:33, 15 February 2021
• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe ...imes <math>t=1</math> and <math>t=2</math> geometrically, as an area under a curve.
  11 KB (2,082 words) - 15:23, 2 January 2022
• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
  51 KB (6,175 words) - 20:58, 6 December 2023
• Modular arithmetic/Introduction
  ...is article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily ...ple, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
  15 KB (2,396 words) - 20:24, 21 February 2024
• 2003 AMC 12A Problems
  {{AMC12 Problems|year=2003|ab=A}} ...B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006 </math>
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2003 AMC 12B Problems
  ...2}{3} \qquad \text {(C) } \frac{2}{3} \qquad \text {(D) } 1 \qquad \text {(E) } \frac{14}{3} ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2005 AIME II Problems/Problem 15
  ...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma pair A = (-5, 12), B = (5, 12), C = (0, 0);
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2004 AIME I Problems/Problem 14
  ...are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> ...= (0,0,0), A=(4*5^.5,-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h);
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 2
  ...The absolute value of the difference between the greatest element of <math>A</math> and the greatest element of <math>B</math> is <math>99</math>. Find ...e median) must be <math>2</math>. Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME II Problems/Problem 7
  ...and <math> n </math> are relatively prime positive integers. Find <math> m+n. </math> ...(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C;
  9 KB (1,501 words) - 05:34, 30 October 2023
• 1993 AIME Problems
  ...etc. If the candidate went <math>\frac{n^{2}}{2}</math> miles on the <math>n^{\mbox{th}}_{}</math> day of this tour, how many miles was he from his star ...many contestants caught <math>n\,</math> fish for various values of <math>n\,</math>.
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1983 AIME Problems/Problem 11
  The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to th triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
  5 KB (865 words) - 21:11, 6 February 2023
• 1983 AIME Problems/Problem 15
  ...rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? pair A=(-0.91,-0.41);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1986 AIME Problems/Problem 9
  pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425)); pair Da=IP(Circle(A,289),A--B),E=IP(Circle(C,324),B--C),Ea=IP(Circle(B,270),B--C);
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1989 AIME Problems/Problem 15
  ...d <math>CPF</math> are drawn with <math>D</math> on <math>BC</math>, <math>E</math> on <math>AC</math>, and <math>F</math> on <math>AB</math> (see the f ...>, and line <math>XP</math> intersects line <math>YZ</math> at point <math>L</math>, then <math>\dfrac{XP}{PL} = \frac{[XPY] + [ZPX]}{[YPZ]}.</math>
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1995 AIME Problems/Problem 12
  ...> where <math>m_{}</math> and <math>n_{}</math> are integers, find <math>m+n.</math> // n = normal to plane
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 3
  ...h> and <math>n</math> are relatively prime positive integers, find <math>m+n.</math> ...math> steps, then it must have gone two steps <tt>N</tt> and two steps <tt>E</tt>, in some permutation. There are <math>\frac{4!}{2!2!} = 6</math> ways
  3 KB (602 words) - 23:15, 16 June 2019
• 1999 AIME Problems/Problem 15
  ...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ..."\(D\)",foot(A,B,C),NE);label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><a
  7 KB (1,169 words) - 15:28, 13 May 2024
• 2001 AIME I Problems/Problem 12
  ...and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2001 AIME I Problems/Problem 6
  ...</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m + n</math>. ...n the diagram below, the lowest <math>y</math>-coordinate at each of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> corresponds to t
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2001 AIME II Problems/Problem 15
  ...<math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. ...ection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2");
  4 KB (518 words) - 15:01, 31 December 2021
• University of South Carolina High School Math Contest/1993 Exam/Problems
  If the width of a particular rectangle is doubled and the length is increased by 3, then the ...B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 6 \qquad \mathrm{(E) \ } 9 </cmath>
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007/Problems
  ...th> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note that <math>[ABC]</math> denotes the area of <math>\triangle A ...>\mathcal{S}</math>, we have that <math>\star (n)=12</math> and <math>0\le n< 10^{7}</math>. If <math>m</math> is the number of elements in <math>\mathc
  8 KB (1,355 words) - 14:54, 21 August 2020
• P versus NP
  .... <math>NP</math> is the class of decision problems that can be solved by a ''non-deterministic'' algorithm in polynomial time. The <math>P</math> ver Since all modern computers (with the exception of a few quantum computers) are deterministic, non-deterministic algorithms are
  6 KB (1,104 words) - 15:11, 25 October 2017
• 2005 AMC 10A Problems/Problem 13
  How many positive integers <math>n</math> satisfy the following condition: <math> (130n)^{50} > n^{100} > 2^{200}\ ?</math>
  2 KB (223 words) - 07:18, 16 July 2022
• 2005 AMC 10A Problems/Problem 25
  ...h> BC = 39</math>, and <math>AC=42</math>. Points <math>D</math> and <math>E</math> are on <math>AB</math> and <math>AC</math> respectively, with <math> ...\textbf{(C) }\frac{1}{3}\qquad \textbf{(D) } \frac{19}{56}\qquad \textbf{(E) } 1 </math>
  6 KB (854 words) - 20:25, 24 July 2022
• 2007 iTest Problems
  A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>
  30 KB (4,794 words) - 23:00, 8 May 2024
• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...ath>, and <math>\triangle CH_AH_B</math> [[concurrence | concur]] at <math>N</math>, the nine-point circle of <math>\triangle ABC</math>.
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2006 Seniors Pancyprian/2nd grade/Problem 4
  ...ibed in a circle, its sides <math>DA</math>, <math>CB</math> meet at <math>E</math> and its sides <math>BA</math>, <math>CD</math> meet at <math>Z</math ...th> intersect the sides of the quadrilateral at the points <math>K , L, M ,N</math> prove that
  856 bytes (129 words) - 00:10, 20 February 2020
• 2020 AMC 10A Problems
  {{AMC10 Problems|year=2020|ab=A}} ...uad\textbf{(C)}\ \frac{7}{12}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}</math>
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Reference
  A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions tha returns the complex number <math>e^{i\theta}</math>, i.e. (cos(theta),sin(theta)) where theta is measured in radians.
  7 KB (1,205 words) - 21:38, 26 March 2024
• Mock AIME 1 Pre 2005 Problems
  ...<math>n</math> are relatively prime positive integers. Determine <math>m + n</math>. <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that <math>AB = BC = 1, CD = 2,</ma
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 3 Pre 2005 Problems
  ...igits are in increasing order. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. (Repeated digits are allowed.) A function <math>f(x)</math> is defined for all real numbers <math>x</math>.
  7 KB (1,135 words) - 23:53, 24 March 2019
• 2007 AIME I Problems/Problem 10
  ...the number of shadings with this property. Find the remainder when <math>N</math> is divided by 1000. ...nother row that is complementary to the first. We remove those two and use a similar argument again to show that every group of <math>6</math> rows can
  13 KB (2,328 words) - 00:12, 29 November 2023
• 2007 Cyprus MO/Lyceum/Problem 22
  ...apezium with <math>\angle A= \angle D=90^\circ</math> and bases <math>AB = a</math> , <math>DC = 2a</math> . If <math>AD = 3a</math> and <math>M</math> ...\ } \frac{5a}{2}\qquad \mathrm{(D) \ } \frac{3a}{\sqrt{3}}\qquad \mathrm{(E) \ } 2a</math>
  979 bytes (166 words) - 02:33, 19 January 2024
• Pascal's Theorem
  pair A=dir(20), B=dir(40), E=dir(210),F=dir(300);
  4 KB (712 words) - 21:57, 25 April 2020
• Circumradius
  ...very triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]]. ==Formula for a Triangle==
  4 KB (729 words) - 16:52, 19 February 2024
• 1998 PMWC Problems
  int triangle(pair z, int n){ for(int i = 0; i < n; ++i){
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2000 PMWC Problems
  <math>y</math> is a number that has <math>8</math> different factors (including the number <mat Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of
  11 KB (1,713 words) - 22:47, 13 July 2023
• Excircle
  ...is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. pair excenter(pair A, pair B, pair C){
  5 KB (843 words) - 03:02, 1 July 2020
• Graph (graph theory)
  ...[[empty set | nonempty]] [[set]] of [[vertex|vertices]] that are joined by a number (possibly zero) of [[edge]]s. Graphs are frequently represented gra ...multiple edges, see [[multigraph]]. If the edges are directed, then <math>E</math> may be defined using ordered pairs from the [[product set]] <math>V
  8 KB (1,428 words) - 10:26, 27 August 2020
• 2007 AMC 10A Problems
  {{AMC10 Problems|year=2007|ab=A}} ...ath>25\%</math> discount. Pam buys 5 tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1959 AHSME Problems
  Each edge of a cube is increased by <math>50</math>%. The percent of increase of the surfa ...xtbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 </math>
  22 KB (3,345 words) - 20:12, 15 February 2023
• 2007 AMC 10A Problems/Problem 18
  ...f its sides has length <math>4</math>, and each two consecutive sides form a right angle. Suppose that <math>\overline{AG}</math> and <math>\overline{CH ..., D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2);
  6 KB (867 words) - 00:17, 20 May 2023
• 2008 USAMO Problems/Problem 3
  (''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y) <cmath>\left|x\right| + \left|y + \frac {1}{2}\right| < n</cmath>
  9 KB (1,585 words) - 01:00, 14 August 2014
• Normal subgroup
  ...he notation <math>\mathbb{Z}/n\mathbb{Z}</math> for the integers mod <math>n</math>.) ...s means <math>aN=Na</math> but it does not imply that for every <math>n\in N</math>, <math>an=na</math>.
  15 KB (2,840 words) - 12:22, 9 April 2019
• Millennium Problems
  ...set of seven problems for which the [[Clay Mathematics Institute]] offered a US \$7 million prize fund (\$1 million per problem) to celebrate the new mi ...24, 2000 at the [[Collège de France]]. [[Timothy Gowers]] first presented a lecture titled ''The Importance of Mathematics'' as an introduction. After
  13 KB (1,969 words) - 17:57, 22 February 2024
• 2005 AMC 8 Problems
  Connie multiplies a number by 2 and gets 60 as her answer. However, she should ...textbf{(B)}\ 15\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 240 </math>
  13 KB (1,821 words) - 22:18, 5 December 2023
• 2004 AMC 12B Problems/Problem 14
  ...<math>AC=5</math>, and <math>BC=12</math>. Points <math>M</math> and <math>N</math> lie on <math>AC</math> and <math>BC</math>, respectively, with <math pair C=(0,0), A=(0,5), B=(12,0), M=(0,4), N=(4,0);
  7 KB (1,083 words) - 22:41, 23 November 2020
• 2009 AMC 10B Problems/Problem 22
  ...pieces as shown in this top view, where <math>M</math> is the midpoint of a top edge. The piece whose top is triangle <math>B</math> contains <math>c</ label("$A$",(-0.4,0.7));
  12 KB (1,868 words) - 03:36, 30 September 2023
• 2000 AMC 8 Problems
  ...\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 21\qquad\mathrm{(E)}\ 37</math> ...uad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2</math>
  15 KB (2,165 words) - 03:32, 13 April 2024
• Radical axis
  The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairl ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B
  10 KB (1,797 words) - 02:05, 24 October 2023
• 1968 AHSME Problems/Problem 35
  MP("a",(5,0),S); MP("J",(0,10),N);
  3 KB (597 words) - 01:52, 16 August 2023
• 2009 USAMO Problems/Problem 5
  pair A = (-.6, .8), B = (.6, .8), C = (.9, -sqrt(.19)), D = (-.9, -sqrt(.19)), G = draw(A--B--C--D--cycle); draw(B--D);
  6 KB (973 words) - 19:24, 18 October 2018
• 2009 IMO Problems/Problem 2
  ...ctively, and let <math>\Gamma</math> be the circle passing through <math>K,L</math> and <math>M</math>. Suppose that the line <math>PQ</math> is tangent dot("O", (50, 38), N);
  2 KB (280 words) - 01:16, 19 November 2023
• 2009 IMO Problems
  ...th>a_i(a_{i+1}-1)</math> for <math>i=1,\ldots,k-1</math>. Prove that <math>n</math> doesn't divide <math>a_k(a_1-1)</math>. ...ctively, and let <math>\Gamma</math> be the circle passing through <math>K,L</math> and <math>M</math>. Suppose that the line <math>PQ</math> is tangent
  3 KB (509 words) - 09:23, 10 September 2020
• 1992 AJHSME Problems
  ...quad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10</math> ...\text{(C)}\ 1\dfrac{3}{12} \qquad \text{(D)}\ 1\dfrac{1}{5} \qquad \text{(E)}\ 1\dfrac{10}{40}</math>
  17 KB (2,346 words) - 13:36, 19 February 2020
• 2010 AMC 12A Problems/Problem 21
  ...xtbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8</math> [Quick note: Since we don't know <math>a</math>, <math>b</math>, and <math>c</math>, we really don't even need the l
  5 KB (969 words) - 21:33, 22 June 2022
• 2010 AMC 12A Problems/Problem 12
  In a magical swamp there are two species of talking amphibians: toads, whose sta Chris: "LeRoy is a frog."
  2 KB (413 words) - 20:08, 31 January 2024
• 2010 AMC 12B Problems
  ...f{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35</math> A big <math>L</math> is formed as shown. What is its area?
  12 KB (1,845 words) - 13:00, 19 February 2020
• Characteristic polynomial
  The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eige ...nomial in <math>t</math>. Here, <math>I</math> refers to the <math>n\times n</math> [[identity matrix]].
  19 KB (3,412 words) - 14:57, 21 September 2022
• Proofs without words
  rt(a^2 - ab + b^2) rt(b^2 - bc + c^2) < rt(a^2 + ac + c^2) int n = 6; pair shiftR = ((n+2),0); real r = 0.3;
  55 KB (7,986 words) - 17:04, 20 December 2018
• 2010 AIME I Problems/Problem 11
  ...<math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. pair A = (8,10), B = (4.5,6.5), C= (9.75,8.25), F=foot(A,B,C), G=2*F-A; fill(A--B--C--cycle,rgb(0.9,0.9,0.9));
  4 KB (636 words) - 16:46, 25 November 2023
• 2007 IMO Problems
  ..._2, \dots , a_n</math> are given. For each <math>i</math> (<math>1\le i\le n</math>) define <cmath>d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}</cmath>
  3 KB (505 words) - 09:24, 10 September 2020
• 2010 AIME II Problems
  {{AIME Problems|year=2010|n=II}} ...ven and no two of whose digits are the same. Find the remainder when <math>N</math> is divided by <math>1000</math>.
  8 KB (1,246 words) - 21:58, 10 August 2020
• 2010 AIME II Problems/Problem 9
  ...<math>n</math> are [[relatively prime]] positive integers. Find <math>m + n</math>. pair A,B,C,D,E,F;
  5 KB (857 words) - 22:22, 27 August 2023
• 1995 AJHSME Problems
  ...t{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 41\% \qquad \text{(E)}\ 59\%</math> ...d \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 22</math>
  14 KB (2,096 words) - 18:29, 2 January 2023
• 2010 AMC 12B Problems/Problem 2
  A big <math>L</math> is formed as shown. What is its area? label("2",(5,1),E);
  802 bytes (110 words) - 10:58, 4 July 2013
• 2010 AMC 12B Problems/Problem 21
  Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that <math>P(1) = P(3) = P(5) = P(7) = a</math>,
  7 KB (1,166 words) - 00:04, 26 February 2024
• 2010 IMO Problems
  ...ft\lfloor a\right\rfloor</math> is greatest integer not greater than <math>a.</math> ...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE
  4 KB (603 words) - 09:22, 10 September 2020
• 2004 AMC 8 Problems
  ...ter length represents <math>72</math> kilometers. How many kilometers does a <math>17</math>-centimeter length represent? ...xtbf{(B)}\ 102\qquad\textbf{(C)}\ 204\qquad\textbf{(D)}\ 864\qquad\textbf{(E)}\ 1224 </math>
  13 KB (1,860 words) - 19:58, 8 May 2023
• 2006 AMC 8 Problems
  ...\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18</math> ...d\textbf{(B)}\ 6\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 26 </math>
  16 KB (2,215 words) - 19:18, 10 April 2024
• 1997 AJHSME Problems
  ...)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997</math> Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is
  12 KB (1,702 words) - 12:35, 6 November 2022
• 1998 AJHSME Problems
  ...d \text{(C)}\ \dfrac{6}{x-1} \qquad \text{(D)}\ \dfrac{x}{6} \qquad \text{(E)}\ \dfrac{x+1}{6}</math> ...text{b}\cdot \text{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>?
  14 KB (1,920 words) - 19:31, 31 January 2024
• 1999 AMC 8 Problems/Problem 15
  ...from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. ...d \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 60</math>
  3 KB (516 words) - 14:50, 21 December 2022
• 2003 AMC 12A Problems/Problem 14
  Points <math>K, L, M,</math> and <math>N</math> lie in the plane of the square <math>ABCD</math> such that <math>AKB pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);
  3 KB (483 words) - 17:26, 22 June 2023
• 2021 AIME II Problems/Problem 5
  ...tau(s)</math> are congruent, is an interval <math>[a,b)</math>. Find <math>a^2+b^2</math>. We start by defining a triangle. The two small sides MUST add to a larger sum than the long side. We are given <math>4</math> and <math>10</ma
  14 KB (2,269 words) - 00:43, 2 January 2023
• 2011 AIME I Problems/Problem 4
  ...overline{AC} </math> at point <math>K</math>. Let <math>M</math> and <math>N</math> be the feet of the perpendiculars from <math>C</math> to <math> \ove pair A,B,C,D,K,L,M,N,P,Q;
  6 KB (1,068 words) - 18:52, 2 August 2023
• 2011 AIME I Problems
  {{AIME Problems|year=2011|n=I}} ...<math>n</math> are relatively prime positive integers, find <math>k + m + n</math>.
  10 KB (1,634 words) - 22:21, 28 December 2023
• 1999 AMC 8 Problems
  ...\text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \qquad \text{(E)}\ \text{None of these}</math> What is the degree measure of the smaller angle formed by the hands of a clock at 10 o clock?
  17 KB (2,394 words) - 19:51, 8 May 2023
• 2000 AMC 8 Problems/Problem 6
  ...s are drawn with the side lengths as labeled. The area of the shaded <math>L</math>-shaped region is pair A,B,C,D;
  2 KB (365 words) - 21:02, 28 July 2023
• 2011 AMC 10B Problems/Problem 9
  pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3)); pair[] ps={A,B,C,D,E};
  4 KB (587 words) - 22:08, 31 August 2023
• 2011 AMC 10B Problems/Problem 16
  A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. W pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqr
  4 KB (606 words) - 06:14, 25 June 2022
• 2011 AMC 10B Problems/Problem 10
  ...extbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)} 101 </math> ...rac{10^{10}}{10^9 + 10^8 + \ldots + 10 + 1}.</cmath> Using the formula for a geometric series, we have <cmath>10^9 + 10^8 + \ldots + 10 + 1 = \dfrac{10^
  2 KB (230 words) - 22:43, 31 August 2023
• Olympiad books
  Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai *''Inequalities: A Mathematical Olympiad Approach'' - '''Radmila Bulajich Manfrino, Jose Anton
  17 KB (2,261 words) - 00:30, 22 April 2024
• Block walking
  ...r right or down. We start off with some easier means of this problem from A to B by labeling closer distances. Let's label these points with letters. label("E", (0,0), SW);
  9 KB (1,228 words) - 21:47, 26 February 2022
• Mock Amc 12
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob ...r more information on Mock AMCs, as well as tips on how to write them, and a complete list of all Mock AMC 10s and Mock AMC 12s.
  18 KB (2,206 words) - 19:41, 24 December 2020
• 2003 AMC 10B Problems/Problem 20
  ...>GC=2</math>. Lines <math>AF</math> and <math>BG</math> intersect at <math>E</math>. Find the area of <math>\triangle AEB</math>. pair A=(0,0), B=(5,0), C=(5,3), D=(0,3);
  5 KB (898 words) - 17:26, 3 July 2023
• 1999 AMC 8 Problems/Problem 4
  for (int a = 0; a < 6; ++a) dot((4*a,3*b));
  1 KB (196 words) - 21:02, 14 April 2017
• 2015 IMO Problems/Problem 4
  ...le of triangle <math>BDF</math> and the segment <math>AB</math>. Let <math>L</math> be the second point of intersection of the circumcircle of triangle ...etric wrt <math>AO</math>, a diameter of <math>\Omega</math> through <math>A</math>. So are <math>F</math> and <math>G</math>, as <math>AF=AG</math>. Th
  3 KB (502 words) - 23:58, 5 October 2015
• Simson line
  In [[geometry]], given a [[triangle]] ABC and a point P on its [[circumcircle]], the three closest points to P on lines AB, Let a triangle <math>\triangle ABC</math> and a point <math>P</math> be given.
  4 KB (638 words) - 06:38, 26 April 2023
• 2003 AMC 8 Problems
  Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of ...thrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22 \qquad\mathrm{(E)}\ 26</math>
  16 KB (2,236 words) - 12:02, 19 February 2024
• 2002 AMC 8 Problems/Problem 10
  ...nd by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and for ( int y = 0; y &l;= 5; ++y )
  2 KB (253 words) - 22:32, 17 February 2024
• 2000 AMC 8 Problems/Problem 25
  ...\overline{BC} </math> and <math> \overline{CD} </math> are joined to form a triangle, the area of that triangle is pair A,B,C,D;
  3 KB (484 words) - 13:59, 22 October 2023
• 1992 AJHSME Problems/Problem 11
  The bar graph shows the results of a survey on color preferences. What percent preferred blue? for (int a = 1; a <= 6; ++a)
  2 KB (226 words) - 00:09, 5 July 2013
• 1992 AJHSME Problems/Problem 21
  Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According for (int a = 1; a <= 6; ++a)
  2 KB (306 words) - 00:10, 5 July 2013
• 2003 AMC 8 Problems/Problem 18
  ...one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);
  2 KB (250 words) - 22:17, 5 January 2024
• 2009 PUMaC Problems
  '''Algebra A''' ...for which all the numbers <cmath>\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots</cmath> are negative.
  25 KB (4,154 words) - 16:27, 2 September 2011
• 1985 AHSME Problems
  ...}16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{(D) \ } 18 \qquad \mathrm{(E) \ }19 </math> In an arcade game, the "monster" is the shaded sector of a circle of radius <math>1</math> cm, as shown in the figure. The missing pie
  17 KB (2,488 words) - 03:26, 20 March 2024
• 2010 AMC 8 Problems
  ...\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 </math> If <math>a @ b = \frac{a\times b}{a+b}</math> for <math>a,b</math> positive integers, then what is <math>5 @ 10</math>?
  18 KB (2,768 words) - 21:05, 9 January 2024
• 2011 AMC 8 Problems
  ...math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? ...}\ \textdollar 2.50 \qquad \textbf{(D) }\ \textdollar 3.00 \qquad \textbf{(E) }\ \textdollar 3.50</math>
  16 KB (2,371 words) - 17:34, 9 January 2024
• Barycentric coordinates
  ...triangle <math> \Delta{A_1}{A_2}{A_3} </math>. These masses then determine a point <math> P </math>, which is the geometric centroid of the three masses The Central NC Math Group published a lecture concerning this topic at https://www.youtube.com/watch?v=KQim7-wrwL
  25 KB (5,067 words) - 22:15, 31 March 2024
• 1989 AHSME Problems
  ...ad\textrm{(B)}\ -2\qquad\textrm{(C)}\ 0\qquad\textrm{(D)}\ 1\qquad\textrm{(E)}\ 57 </math> ...qquad\textrm{(C)}\ \frac{2}7\qquad\textrm{(D)}\ \frac{5}{12}\qquad\textrm{(E)}\ \frac{7}{12} </math>
  15 KB (2,247 words) - 13:44, 19 February 2020
• 2012 AMC 10A Problems/Problem 15
  pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); draw(A--(2,0)); draw((0,-1)--(2,-1)); draw((0,-2)--(1,-2));
  6 KB (935 words) - 23:41, 13 September 2023
• 2012 AMC 12A Problems/Problem 22
  Distinct planes <math>p_1,p_2,....,p_k</math> intersect the interior of a cube <math>Q</math>. Let <math>S</math> be the union of the faces of <math> ...\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 </math>
  5 KB (940 words) - 17:13, 4 April 2020
• 2012 AMC 12B Problems/Problem 17
  ...}5\qquad\textbf{(C) }\frac{32}5\qquad\textbf{(D) }\frac{33}5\qquad\textbf{(E) }\frac{34}5 </math> ...; pair DA = extension(D,A,(0,-2),(1,-2)); draw(A--AB--B--BC--C--CD--D--DA--A, dotted); dot(AB^^BC^^CD^^DA);</asy>
  12 KB (2,183 words) - 21:05, 23 December 2023
• 2012 AMC 12B Problems/Problem 21
  ...\ 20\sqrt{3}+16 \qquad\textbf{(D)}\ 20\sqrt{2}+13 \sqrt{3} \qquad\textbf{(E)}\ 21\sqrt{6}</math> pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60);
  9 KB (1,451 words) - 10:01, 27 December 2022
• 2015 AIME I Problems
  {{AIME Problems|year=2015|n=I}} ...successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>.
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  {{AIME Problems|year=2012|n=I}} ...s with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 7
  At each of the sixteen circles in the network below stands a student. A total of <math>3360</math> coins are distributed among the sixteen students pair A = (0,0);
  6 KB (1,058 words) - 01:49, 25 November 2023
• 2012 AIME I Problems/Problem 13
  ...and <math>d</math> is not divisible by the square of any prime. Find <math>a+b+c+d.</math> ...ath> A <math>60^\circ</math> counter-clockwise rotation about vertex <math>A</math> maps <math>X</math> to <math>X'</math> and <math>B</math> to <math>C
  11 KB (1,889 words) - 20:42, 25 January 2023
• 2007 iTest Problems/Problem 15
  Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and ...one of its sides coincides with a side of the square. Then "circumscribe" a circle around the
  3 KB (539 words) - 21:01, 29 July 2018
• 2010 AMC 8 Problems/Problem 3
  ....4), G=(1,-5.2), H=(1,-5), J=(1,-4.8), K=(1,-4.6), L=(1,-4.4), M=(1,-4.2), N=(1,-4), P=(1,-3.8), Q=(1,-3.6), R=(1,-3.4), S=(1,-3.2), T=(1,-3), U=(1,-2.8 ...D(B--(8,-2),linewidth(0.4)); D(B--S_1); D(T--R_1); D(N--Q_1); D(H--P_1); D(A--O_1); D(C_1--E_1); D(E_1--F_1); D(F_1--D_1); D(D_1--C_1); D(G_1--L_2); D(L
  5 KB (847 words) - 19:59, 5 February 2023
• 2006 SMT/Geometry Problems/Problem 4
  ...ath> and <math> X </math> is on the same side of line <math> AB </math> as a given point <math> C </math>. label("$A$",(-1,0),SW);
  4 KB (622 words) - 15:07, 28 May 2012
• 1952 AHSME Problems
  If the radius of a circle is a rational number, its area is given by a number which is: ...xt{integral} \qquad \textbf{(D)\ } \text{a perfect square }\qquad \textbf{(E)\ } \text{none of these} </math>
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1980 AHSME Problems
  ...text{(B)} \ 13 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ 16</math> The degree of <math>(x^2+1)^4 (x^3+1)^3</math> as a polynomial in <math>x</math> is
  15 KB (2,302 words) - 10:47, 30 April 2021
• 1998 USAMO Problems/Problem 6
  ...A_{i}A_{i+1}A_{i+2}A_{i+3}</math> have an inscribed circle. (Here <math>A_{n+j} = A_{j}</math>.) ...ath> and <math>A_{i+2}A_{i+3}A_{i+4}A_{i+5}</math> in an equiangular <math>n</math>-gon are tangential, and <math>A_iA_{i+3}</math> is the longest side
  5 KB (871 words) - 18:59, 10 May 2023
• 2012 AMC 8 Problems
  ...many pounds of meat does she need to make <math> 24 </math> hamburgers for a neighborhood picnic? ...C)}\hspace{.05in}7\frac{1}2\qquad\textbf{(D)}\hspace{.05in}8\qquad\textbf{(E)}\hspace{.05in}9 </math>
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2012 AMC 8 Problems/Problem 5
  pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0);
  2 KB (208 words) - 20:16, 3 January 2024
• 2013 AMC 12A Problems
  {{AMC12 Problems|year=2013|ab=A}} Square <math> ABCD </math> has side length <math> 10 </math>. Point <math> E </math> is on <math> \overline{BC} </math>, and the area of <math> \bigtria
  14 KB (2,206 words) - 19:31, 15 May 2024
• 2013 AMC 12A Problems/Problem 11
  Triangle <math>ABC</math> is equilateral with <math>AB=1</math>. Points <math>E</math> and <math>G</math> are on <math>\overline{AC}</math> and points <mat real s=1/2,m=5/6,l=1;
  2 KB (322 words) - 20:12, 24 November 2020
• 2013 AIME II Problems
  {{AIME Problems|year=2013|n=II}} ...\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>.
  8 KB (1,402 words) - 12:17, 13 March 2020
• 2013 AIME II Problems/Problem 13
  ...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. After drawing the figure, we suppose <math>BD=a</math>, so that <math>CD=3a</math>, <math>AC=4a</math>, and <math>AE=ED=b</
  13 KB (2,116 words) - 23:24, 21 March 2024
• 2003 AMC 12B Problems/Problem 10
  pair A = (5,0), B = A+10*dir(72), C = (0,r+h), E = (-5,0), D = E+10*dir(108); draw(A--B--C--D--E--cycle);
  1 KB (237 words) - 23:06, 3 February 2020
• 2014 AIME II Problems/Problem 11
  ...me positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. ...c{\sqrt{3}}{2}</math>, so <math>D\left(\frac{1}{2}, 0\right)</math>, <math>E\left(-\frac{\sqrt{3}}{2}, 0\right)</math>, and <math>R\left(0, \frac{\sqrt{
  6 KB (1,059 words) - 18:24, 20 January 2024
• 2014 AIME II Problems/Problem 14
  ...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...H = MP("H",foot(A,B,C),N), N = MP("N",0.5(H+M),NE), P = MP("P",IP(A--D,L(N,N-(1,1),0,10)));
  11 KB (1,442 words) - 19:28, 21 October 2023
• 2013 IMO Problems/Problem 4
  ...M</math>, and let <math>Y</math> be the point such that <math>WY</math> is a diameter of <math>\omega_2</math>. Prove that <math>X, Y</math> and <math>H Draw a good diagram, or use the one below. What do you notice? (In particular, wha
  11 KB (1,991 words) - 01:31, 19 November 2023
• 2016 AMC 10A Problems
  {{AMC10 Problems|year=2016|ab=A}} ...xtbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math>
  14 KB (2,104 words) - 22:26, 16 September 2022
• 1969 IMO Problems/Problem 5
  ...ane such that no three are collinear, prove that there are at least <math>(n-3)/2</math> convex quadrilaterals whose vertices are four of the given poin ...DAC would be less than BAC - contradiction. So there are none of the other n-3 points inside the area of ABC.
  2 KB (394 words) - 13:44, 29 January 2021
• 1971 IMO Problems/Problem 5
  ...th>m</math> points in <math>S</math> which are at unit distance from <math>A</math>. ...ean distance between the points is <math>1</math>) and this will follow as a consequence.
  4 KB (749 words) - 14:09, 29 January 2021
• 2013 AMC 8 Problems
  ...xtbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math> ...lf-pound packages for just \$3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk)
  15 KB (2,162 words) - 20:05, 8 May 2023
• 2013 AMC 8 Problems/Problem 18
  Isabella uses one-foot cubical blocks to build a rectangular fort that is <math>12</math> feet long, <math>10</math> feet wi triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P;
  3 KB (412 words) - 22:30, 18 December 2023
• 1994 AHSME Problems
  ...36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\ 1296^{26} </math> A large rectangle is partitioned into four rectangles by two segments paralle
  14 KB (2,124 words) - 13:39, 19 February 2020
• 1981 AHSME Problems
  ...uad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 </math> Point <math>E</math> is on side <math>AB</math> of square <math>ABCD</math>. If <math>EB<
  17 KB (2,633 words) - 15:44, 16 September 2023
• 1968 AHSME Problems
  Let <math>P</math> units be the increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter ...(B) } \pi\quad\text{(C) } \frac{\pi^2}{2}\quad\text{(D) } \pi^2\quad\text{(E) } 2\pi</math>
  16 KB (2,571 words) - 14:13, 20 February 2020
• 2014 AMC 10A Problems
  {{AMC10 Problems|year=2014|ab=A}} ...d\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}\ \frac{170}{3}\qquad\textbf{(E)}\ 170</math>
  15 KB (2,190 words) - 15:21, 22 December 2020
• 2014 AMC 10A Problems/Problem 13
  pair A = rotate(60,B)*C; pair E = rotate(270,A)*B;
  7 KB (1,040 words) - 02:07, 8 October 2022
• 2014 AMC 10A Problems/Problem 14
  ...two perpendicular lines intersecting at the point <math>A(6,8)</math> have a sum of zero. What is the area of <math>\triangle APQ</math>? ...\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 </math>
  6 KB (1,001 words) - 13:07, 25 July 2022
• 2014 AMC 10A Problems/Problem 16
  ...<math>ABCD</math>, <math>AB=1</math>, <math>BC=2</math>, and points <math>E</math>, <math>F</math>, and <math>G</math> are midpoints of <math>\overline pair E = (1,1);
  9 KB (1,411 words) - 19:51, 25 July 2023
• 2014 AMC 10A Problems/Problem 17
  ...quad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 </math> ...(2, 2, 4); (3, 3, 6)</math> have <math>3</math> ways to rearrange them for a total of <math>9</math> ways. Adding them up, we get <math>45</math> ways.
  3 KB (496 words) - 22:43, 21 November 2022
• 2014 USAJMO Problems/Problem 2
  ...riangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^{\circ}</math>, and let <math>O</math> and <math>H</math> denote the cir (a) Prove that line <math>OH</math> intersects both segments <math>AB</math> a
  7 KB (1,197 words) - 19:40, 28 February 2023
• 2018 AIME I Problems/Problem 14
  ...sequences of jumps of no more than <math>12</math> jumps that end at <math>E</math>. ...arrow P_3 \leftrightarrow E</math>. We can count the number of left/right (L/R) paths of length <math>\le 11</math> that start at <math>S</math> and end
  4 KB (649 words) - 17:29, 22 December 2023
• 2014 AMC 12B Problems/Problem 21
  In the figure, <math> ABCD </math> is a square of side length <math> 1 </math>. The rectangles <math> JKHG </math> pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0
  8 KB (1,307 words) - 02:07, 1 January 2023
• 2014 AIME I Problems/Problem 13
  ...nts <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, pair A = (0,sqrt(850));
  9 KB (1,404 words) - 21:07, 13 October 2023
• 1969 AHSME Problems
  <math>\frac{a}{b},a \ne b,b \ne 0</math>, the value of the fraction is changed to <math>\frac{c <math>\text{(A) } \frac{1}{c-d}\quad
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1988 AHSME Problems
  <math>\text{(A)}\ \sqrt{20} \qquad \text{(E)}\ 2\sqrt{13} </math>
  17 KB (2,535 words) - 13:45, 19 February 2020
• 1986 AHSME Problems
  <math>\textbf{(A)}\ 2y \qquad \textbf{(E)}\ 0 </math>
  17 KB (2,512 words) - 18:30, 12 October 2023
• 2007 iTest Problems/Problem 14
  ...ath>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>? <math>\text{(A) } 0 \quad
  2 KB (265 words) - 21:23, 4 February 2023
• 1976 AHSME Problems
  <math>\textbf{(A) }-2\qquad \textbf{(E) }3 </math>
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1972 AHSME Problems
  <math>\text{(A) I and II}\qquad \text{(E) I, II, and IV }</math>
  17 KB (2,732 words) - 13:54, 20 February 2020
• 2007 iTest Problems/Problem 19
  ...rechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!" ...son a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.
  3 KB (397 words) - 01:18, 29 October 2023
• 2007 iTest Problems/Problem 18
  ...ic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers. <math>\text{(A) }0\qquad
  2 KB (393 words) - 17:01, 10 June 2018
• 2007 iTest Problems/Problem 17
  <math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad <math>\text{(E) }1\qquad
  2 KB (266 words) - 21:30, 4 February 2023
• 2007 iTest Problems/Problem 16
  <math>\text{(A) }1\qquad \text{(E) }41\qquad
  3 KB (420 words) - 20:28, 17 June 2018
• 2007 iTest Problems/Problem 21
  James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's. <math>\text{(A) }0\qquad
  2 KB (247 words) - 03:34, 14 June 2018
• 2007 iTest Problems/Problem 22
  <math>\text{(A) } 0 \quad \text{(E) } 4 \quad
  929 bytes (137 words) - 22:05, 10 January 2019
• 2007 iTest Problems/Problem 25
  <math>\text{(A) } 0\qquad \text{(E) } 4\qquad
  2 KB (270 words) - 14:35, 29 July 2018
• 2007 iTest Problems/Problem 23
  <math>\text{(A) } 0\quad \text{(E) } -2\quad
  1 KB (202 words) - 16:48, 24 November 2018
• 2007 iTest Problems/Problem 24
  ...such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube. Find the remainder when <math>N</math> is divided by <math>25</math>.
  2 KB (340 words) - 19:49, 30 June 2018
• 2007 IMO Problems/Problem 4
  ...h>BC</math> is <math>K</math> and the midpoint of <math>AC</math> is <math>L</math>. Prove that the triangles <math>RPK</math> and <math>RQL</math> have real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
  8 KB (1,480 words) - 14:52, 5 August 2022
• 2014 IMO Problems/Problem 3
  ...}</math>. Point <math>H</math> is the foot of the perpendicular from <math>A</math> to <math>BD</math>. Points <math>S</math> and <math>T</math> lie on pair A = (0,1), C=(0,-1);
  6 KB (1,071 words) - 03:58, 8 September 2018
• 2014 IMO Problems/Problem 4
  ...and <math>\angle{CAQ}=\angle{ABC}</math>. Points <math>M</math> and <math>N</math> lie on lines <math>AP</math> and <math>AQ</math>, respectively, such label("$A$", (1.880000000000002,3.760000000000004), NE * labelscalefactor);
  13 KB (1,803 words) - 12:00, 19 May 2024
• 1965 AHSME Problems
  <math>\textbf{(A)}\ 0 \qquad \textbf{(E) }\ \text{more than 4} </math>
  19 KB (2,873 words) - 18:57, 16 August 2023
• 1963 AHSME Problems
  <math>\textbf{(A)}\ (0,0)\qquad \textbf{(E)}\ (-2,2) </math>
  20 KB (3,122 words) - 14:17, 20 February 2020
• 2008 UNCO Math Contest II Problems
  Let <math>S = {a,b,c,d}</math> be a set of four positive integers. If pairs of distinct elements of <math>S</ma ...btained: <math>5, 10, 11, 13, 14, 19.</math> Determine the values of <math>a,
  6 KB (871 words) - 22:13, 7 November 2014
• 2008 UNCO Math Contest II Problems/Problem 6
  Points <math>A</math> and <math>B</math> are on the same side of line <math>L</math> in the plane. <math>A</math> is <math>5</math> units away
  2 KB (252 words) - 22:28, 18 March 2018
• 1988 AHSME Problems/Problem 9
  ...), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N);
  2 KB (367 words) - 08:24, 12 July 2022
• 1987 AHSME Problems/Problem 13
  ...nto a roll for cash registers by wrapping it <math>600</math> times around a cardboard tube of diameter <math>2</math> cm, forming a roll <math>10</math> cm in diameter. Approximate the length of the paper in
  1 KB (199 words) - 14:24, 1 March 2018
• 2016 AMC 12A Problems
  {{AMC12 Problems|year=2016|ab=A}} ...xtbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math>
  15 KB (2,348 words) - 17:20, 19 January 2024
• 2014 AMC 8 Problems
  ...quad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10</math> Paul owes Paula 35 cents and has a pocket full of 5-cent coins, 10-cent coins, and 25-cent coins that he can u
  13 KB (1,957 words) - 12:08, 13 January 2024
• 2015 AMC 10A Problems/Problem 19
  ...th>\angle ACB</math> intersect <math>AB</math> at <math>D</math> and <math>E</math>. What is the area of <math>\bigtriangleup CDE</math>? ...ac{15\sqrt{3}}{8}\qquad\textbf{(D) }\dfrac{50-25\sqrt{3}}{2}\qquad\textbf{(E) }\dfrac{25}{6} </math>
  9 KB (1,513 words) - 23:44, 13 May 2024
• 2015 AMC 12B Problems
  ...extbf{(C)}\; \dfrac{7}{4} \qquad\textbf{(D)}\; \dfrac{9}{4} \qquad\textbf{(E)}\; 6</math> Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishe
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AIME I Problems/Problem 7
  ...and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</m pair A,B,C,D,E,F,G,H,J,K,L,M,N;
  6 KB (1,105 words) - 21:02, 9 November 2023
• 2015 AIME II Problems
  {{AIME Problems|year=2015|n=II}} .../math> percent greater than another integer. Find the remainder when <math>N</math> is divided by <math>1000</math>.
  8 KB (1,326 words) - 19:15, 13 January 2024
• 1998 IMO Problems/Problem 3
  For any positive integer <math>n</math>, let <math>d(n)</math> denote the number of positive divisors of <math>n</math> (including 1 and <math>n</math> itself). Determine all positive integers <math>k</math> such that
  3 KB (566 words) - 23:48, 18 November 2023
• 2015 AIME II Problems/Problem 7
  ...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. so the answer is <math>m + n = 36 + 125 = \boxed{161}</math>.
  7 KB (1,180 words) - 14:08, 14 February 2023
• 2015 AIME II Problems/Problem 15
  ...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);
  31 KB (5,086 words) - 19:15, 20 December 2023
• 2016 AIME I Problems
  {{AIME Problems|year=2016|n=I}} ...and <math>1</math> satisfy <math>S(a)S(-a)=2016</math>. Find <math>S(a)+S(-a)</math>.
  8 KB (1,360 words) - 12:19, 29 January 2022
• 2016 AMC 10A Problems/Problem 24
  A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}</math>. Three of the sides of this quadr ...\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500</math>
  24 KB (3,861 words) - 14:17, 26 April 2024
• 2015 USAMO Problems
  ...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle. ...math>S</math> is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set <math>T \subseteq S</ma
  4 KB (629 words) - 13:49, 22 November 2023
• Aops font
  The AoPS font is a font that can be used to make certain special symbols in the forum. You enc | <span class="aops-font">a</span>
  6 KB (924 words) - 21:31, 22 February 2024
• 1997 JBMO Problems
  ...y 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than <math>\frac{1}{8}</math>. <cmath> E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. </cmath>
  2 KB (245 words) - 22:44, 14 September 2023
• 2016 AMC 12A Problems/Problem 15
  ...vely, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math ...}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math>
  8 KB (1,255 words) - 09:05, 5 September 2022
• 2016 AMC 10B Problems/Problem 19
  ...<math>ABCD</math> has <math>AB=5</math> and <math>BC=4</math>. Point <math>E</math> lies on <math>\overline{AB}</math> so that <math>EB=1</math>, point label("$Q$",(3.07692307692,2.15384615384),N);
  11 KB (1,608 words) - 03:18, 23 January 2023
• 2016 AMC 10B Problems/Problem 8
  <math>\textbf{(A)}\ 0 \qquad \textbf{(E)}\ 8</math>
  2 KB (366 words) - 12:46, 2 July 2023
• 2016 AMC 10B Problems/Problem 22
  ...</math>, <math>B</math> beat <math>C</math>, and <math>C</math> beat <math>A?</math> <math>\textbf{(A)}\ 385 \qquad
  6 KB (1,034 words) - 00:30, 31 December 2022
• 2016 AIME I Problems/Problem 6
  ...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
  14 KB (2,397 words) - 20:04, 27 August 2023
• 2016 AIME I Problems/Problem 3
  ...r of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no ...5,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A;
  4 KB (636 words) - 13:00, 22 December 2020
• 2016 AIME I Problems/Problem 9
  Triangle <math>ABC</math> has <math>AB=40,AC=31,</math> and <math>\sin{A}=\frac{1}{5}</math>. This triangle is inscribed in rectangle <math>AQRS</ma ...hen <math>m=n</math>. Thus the maximum possible value of <math>\cos(m)\cos(n)</math> is <math>\frac{1}{2}(\frac{1}{5}+1)=\frac{3}{5}</math> so the maxim
  9 KB (1,526 words) - 02:31, 29 December 2021
• 2016 AIME II Problems/Problem 7
  ...n a side of <math>ABCD</math> and each vertex of <math>EFGH</math> lies on a side of <math>IJKL</math>. Find the difference between the largest and smal ...math>12^{2} \cdot 14, 11 \cdot 12 \cdot 14, 11^{2} \cdot 14</math> forming a geometric progression).
  2 KB (316 words) - 00:10, 2 September 2021
• 2016 AIME II Problems/Problem 14
  ...planes). There is a point <math>O</math> whose distance from each of <math>A,B,C,P,</math> and <math>Q</math> is <math>d</math>. Find <math>d</math>. ...ngle ABC</math>, and <math>L</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. We must have <math>\tan(\angle KLP+ \angle QLK)=
  15 KB (2,560 words) - 01:44, 1 July 2023
• 2006 iTest Problems
  <math>\mathrm{(A)}\, 8</math> <math>\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}</math>
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2016 USAMO Problems/Problem 5
  ...<math>M\in\overline{AB},</math> <math>Q\in\overline{AC},</math> and <math>N, P\in\overline{BC}.</math> Let <math>S</math> be the intersection of lines ...BC</math>. Let <math>R</math> be the point such that <math>NPQR</math> is a rhombus. It follows that <math>OD\perp QR</math>.
  8 KB (1,449 words) - 00:09, 12 October 2023
• Mock AIME 3 2010 Problems
  -1. Find the sum of all distinct integers n such that n|243. ...can distribute 1 piece of candy among 243 hungry schoolchildren sitting in a row
  14 KB (2,904 words) - 18:24, 16 May 2017
• 2017 AMC 12A Problems/Problem 21
  ...ath>a_{n}x^n + a_{n-1}x^{n-1} + \dots + a_{1}x + a_0</math> for some <math>n\geq{1}</math>, all of whose coefficients <math>a_i</math> are elements of < <math> \textbf{(A)}\ 4
  2 KB (372 words) - 10:48, 27 March 2023
• 2017 AMC 10B Problems/Problem 14
  ...N \leq 2020</math> . What is the probability that the remainder when <math>N^{16}</math> is divided by <math>5</math> is <math>1</math>? ...quad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{4}{5}\qquad\textbf{(E)}\ 1</math>
  4 KB (665 words) - 01:17, 4 December 2023
• 2017 AMC 10B Problems/Problem 21
  ...qquad\textbf{(C)}\ 2\sqrt{2}\qquad\textbf{(D)}\ \frac{17}{6}\qquad\textbf{(E)}\ 3</math> ...>AD = BD = CD = 5</math>, and <math>[ADB] = [ADC] = 12</math>. Since <math>A = rs,</math> we have <math>r = \frac As</math>, so the inradius of <math>\t
  4 KB (601 words) - 00:34, 8 August 2023
• 2017 USAMO Problems/Problem 3
  ...<math>\triangle MAN</math> intersect at points <math>L_1</math> and <math>L.</math> Prove that <math>\Omega</math> passes through the midpoint of eithe ...the orthocenter of <math>\triangle DMX \implies</math> the points <math>X, A,</math> and <math>S</math> are collinear.
  4 KB (711 words) - 18:24, 8 May 2023
• 2017 UNCO Math Contest II Problems
  A prime is an integer strictly greater than one that is evenly divisible by n pair A=24*dir(40),B=15*dir(220);
  7 KB (1,192 words) - 15:14, 20 August 2020
• 2017 UNCO Math Contest II Problems/Problem 2
  pair A=dir(60),B=dir(120),C=dir(180),D=dir(240),E=dir(300),F=dir(360),O=(0,0); ...sqrt(3))*A,H=(2/sqrt(3))*B,I=(2/sqrt(3))*C,J=(2/sqrt(3))*D,K=(2/sqrt(3))*E,L=(2/sqrt(3))*F;
  1 KB (195 words) - 18:22, 16 January 2023
• 1972 AHSME Problems/Problem 30
  label("L",(3.75,h/2),W); ...t one corner touches the opposite side. The length in inches of the crease L in terms of angle <math>\theta</math> is
  3 KB (431 words) - 19:52, 23 June 2021
• 2020 AMC 12B Problems
  ...\sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}</math> ...) } \frac{4780}{4779} \qquad \textbf{(D) } \frac{108}{107} \qquad \textbf{(E) } \frac{81}{80} </math>
  16 KB (2,417 words) - 01:03, 28 April 2022
• 2018 AMC 10A Problems/Problem 23
  ...h> units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like th pair A, B, C, D, F;
  16 KB (2,454 words) - 13:30, 23 September 2023
• 2018 AMC 12A Problems/Problem 23
  ...> respectively, so that <math>PU=AG=1.</math> Let <math>M</math> and <math>N</math> be the midpoints of segments <math>\overline{PA}</math> and <math>\o <math>\textbf{(A) } 76 \qquad
  12 KB (1,878 words) - 22:11, 23 October 2021
• 1977 AHSME Problems/Problem 19
  Let <math>E</math> be the point of intersection of the diagonals of convex quadrilatera <math>\textbf{(A) }PQRS\text{ is a parallelogram}\\
  2 KB (305 words) - 08:38, 28 June 2018
• 2019 IMO Problems
  ''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>'' ...at <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly b
  4 KB (713 words) - 08:59, 10 September 2020
• 2002 Indonesia MO Problems
  ...th>n^4 - n^2</math> is divisible by <math>12</math> for any integers <math>n > 1</math>. <math>\left\{\begin{array}{l}x+y+z = 6\\x^2 + y^2 + z^2 = 12\\x^3 + y^3 + z^3 = 24\end{array}\right.</ma
  2 KB (386 words) - 00:08, 4 August 2018
• 2013 Indonesia MO Problems
  In a <math>4 \times 6</math> grid, all edges and diagonals are drawn (see attach ...ines <math>ME, MF</math> intersects <math>\omega</math> at points <math>K, L</math> respectively. Prove that <math>AM, BL, CK</math> are concurrent.
  4 KB (648 words) - 00:29, 13 August 2018
• 2009 Indonesia MO Problems
  Find all positive integers <math> n\in\{1,2,3,\ldots,2009\}</math> such that <cmath> 4n^6 + n^3 + 5</cmath>
  3 KB (565 words) - 23:33, 26 August 2018
• 2014 Indonesia MO Problems
  ...1,2,\ldots,9</math> once each such that the sum of any two numbers sharing a side is prime? ...gral solution <math>(x,y)</math>. Determine all possible values of <math>m-n</math>.
  3 KB (540 words) - 23:40, 26 August 2018
• 2006 iTest Problems/Problem 18
  <math>\text{(A) Ptolemy; Archimedes}\qquad \text{(E) Goldbach; Bernoulli}\qquad
  2 KB (252 words) - 12:04, 27 November 2018
• 2006 iTest Problems/Problem 15
  <math>\text{(A) }37\qquad \text{(E) }41\qquad
  1 KB (197 words) - 23:31, 27 November 2018
• 2006 iTest Problems/Problem 16
  ...wins four games) will require the full seven games to determine a winner? <math>\text{(A) }\frac{1}{16}\qquad
  2 KB (277 words) - 13:03, 29 November 2018
• 2006 iTest Problems/Problem 17
  <math>\text{(A) }\frac{2305}{2401}\qquad \text{(E) }\frac{1}{2401}\qquad
  2 KB (290 words) - 18:27, 30 November 2018
• 2006 iTest Problems/Problem 14
  ...</math> is the smallest positive integer such that <math>2^x</math> leaves a remainder of <math>1</math> when divided by <math>5</math>, <math>7</math>, <math>\text{(A) } 15 \quad
  2 KB (238 words) - 20:27, 30 November 2018
• 2019 AIME II Problems/Problem 7
  pair A, B, C, D, E, F, G, H, I, J, K, L; A = intersectionpoints(Circle(B,120),Circle(C,180))[0];
  7 KB (1,053 words) - 14:58, 14 January 2024
• 2019 AIME II Problems/Problem 15
  ...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. pair A,B,C,P,Q,X,Y,O;
  7 KB (1,115 words) - 03:11, 7 January 2024
• 1988 IMO Problems
  ...s 0 and 1 in such a manner that each of the sets has exactly <math> \frac {n}{2}</math> zeros? A function <math> f</math> defined on the positive integers (and taking posit
  3 KB (499 words) - 13:29, 2 August 2021
• 2020 AMC 10B Problems
  ...textbf{(B)}\ -3 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 21</math> ...xtbf{(B)}\ 25 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 45</math>
  16 KB (2,497 words) - 21:14, 12 November 2023
• 2019 USAJMO Problems
  ...with an asterisk <math>(*)</math>, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement ...owls contains an apple, and each of the last <math>b</math> bowls contains a pear.
  4 KB (652 words) - 14:21, 10 March 2024
• 2019 USAMO Problems
  ...with an asterisk <math>(*)</math>, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement ...es}}(n)\ldots))=\frac{n^2}{f(f(n))}</cmath>for all positive integers <math>n</math>. Given this information, determine all possible values of <math>f(10
  3 KB (495 words) - 13:47, 22 November 2023
• 2019 AMC 10A Problems/Problem 14
  ...two or more of the lines. What is the sum of all possible values of <math>N</math>? ...f{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21</math>
  4 KB (718 words) - 18:32, 30 October 2022
• 2019 AIME I Problems
  {{AIME Problems|year=2019|n=I}} ...99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.
  8 KB (1,331 words) - 06:57, 4 January 2021
• 2019 AMC 12B Problems
  ...textbf{(C) } \frac{7}{8} \qquad \textbf{(D) } \frac{9}{10} \qquad \textbf{(E) } \frac{11}{12}</math> ...h>n-2</math> is prime." Which of the following values of <math>n</math> is a counterexample to this statement?
  16 KB (2,477 words) - 15:41, 9 September 2023
• 2019 AMC 12B Problems/Problem 10
  ...ing along any portion of a road more than once. (Paula is allowed to visit a city more than once.) pair A = (j,i);
  8 KB (1,306 words) - 22:18, 1 February 2020
• 2019 AMC 12B Problems/Problem 13
  ...1,2,3....</math> What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?<br> ...ad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}</math>
  11 KB (1,795 words) - 00:24, 26 March 2023
• 2019 AIME I Problems/Problem 6
  ...h>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal ...<math>\angle LNK=\beta</math>. Let <math>P</math> be the project of <math>L</math> onto line <math>NK</math>. Note <math>\angle KLP=\beta</math>.
  11 KB (1,717 words) - 20:11, 19 January 2024
• 1988 IMO Problems/Problem 5
  ...ath> AB, AC</math> at the points <math> K,L</math> respectively. If <math> E</math> and <math> E_1</math> dnote the areas of triangles <math> ABC</math> <cmath> \frac {E}{E_1} \geq 2.</cmath>
  3 KB (578 words) - 11:38, 30 January 2021
• 2006 iTest Problems/Problem 20
  <math>\textbf{(A) }2\qquad \textbf{(E) }9\qquad
  4 KB (334 words) - 12:49, 15 January 2020
• 2006 iTest Problems/Problem 19
  <math>\textbf{(A) }7\qquad \textbf{(E) }11\qquad
  4 KB (332 words) - 00:10, 27 January 2023
• 2020 AIME I Problems/Problem 15
  ...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> The following is a power of a point solution to this menace of a problem:
  16 KB (2,678 words) - 22:45, 27 November 2023
• 2006 Indonesia MO Problems/Problem 5
  ...ath> G</math> is the centroid of triangle <math> ABC</math>. A line <math> l</math> passes through <math> G</math>, intersecting line <math> AB</math> a pair B=(0,0),A=(20,90),C=(100,0),M=(50,0),G=(40,30);
  3 KB (489 words) - 18:48, 14 December 2019
• 2019 IMO Problems/Problem 5
  ...T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation: ...herwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> wou
  10 KB (1,760 words) - 01:51, 19 November 2023
• 2020 AMC 10A Problems/Problem 10
  ...6</math>, and <math>343</math> cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except ...xtbf{(B)}\ 658\qquad\textbf{(C)}\ 664\qquad\textbf{(D)}\ 720\qquad\textbf{(E)}\ 749 </math>
  5 KB (846 words) - 21:39, 10 November 2023
• 2020 AMC 10A Problems/Problem 19
  .... How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and mo pair A = (0.082, 0.378);
  8 KB (1,269 words) - 05:36, 4 November 2022
• 2020 AMC 12A Problems/Problem 25
  The number <math>a=\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively <cmath> \lfloor x \rfloor \cdot \{x\} = a \cdot x^2</cmath>
  9 KB (1,541 words) - 17:13, 21 July 2022
• 2020 AMC 10B Problems/Problem 23
  ...<math>ABCD</math> in the coordinate plane has vertices at the points <math>A(1,1), B(-1,1), C(-1,-1),</math> and <math>D(1,-1).</math> Consider the foll <math>\quad\bullet\qquad</math> <math>L,</math> a rotation of <math>90^{\circ}</math> counterclockwise around the origin;
  10 KB (1,653 words) - 23:33, 3 August 2023
• 2020 AMC 10B Problems/Problem 24
  ...integers <math>n</math> satisfy <cmath>\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?</cmath>(Recall that <math>\lfloor x\rfloor</math> is the greatest ...textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32</math>
  10 KB (1,550 words) - 10:26, 9 March 2024
• 2020 AMC 10B Problems/Problem 21
  In square <math>ABCD</math>, points <math>E</math> and <math>H</math> lie on <math>\overline{AB}</math> and <math>\over pair A, B, C, D, E, F, G, H, I, J;
  11 KB (1,850 words) - 23:33, 27 March 2024
• 2020 AMC 12B Problems/Problem 5
  ...</math> more games than team <math>A.</math> How many games has team <math>A</math> played? ...f{(B) } 27 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 63</math>
  5 KB (783 words) - 15:01, 8 June 2023
• 2022 AMC 10A Problems
  {{AMC10 Problems|year=2022|ab=A}} ...textbf{(C)}\ \frac{33}{10}\qquad\textbf{(D)}\ \frac{109}{33}\qquad\textbf{(E)}\ \frac{15}{4}</math>
  18 KB (2,662 words) - 02:08, 9 March 2024
• 2020 FMC 12A Problems
  ...identical bottles. Given that Amy spent <math>3</math> more dollars to buy a standard pack of soda in <math>2020</math> than in <math>2019, </math> what ...(B)}\ 1.65 \qquad\textbf{(C)}\ 1.7 \qquad\textbf{(D)}\ 1.75 \qquad\textbf{(E)}\ 1.8</math>
  15 KB (2,388 words) - 13:24, 9 June 2020
• 2019 IMO
  integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>'' ...at <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly b
  4 KB (704 words) - 00:54, 9 July 2021
• 2021 AIME I Problems/Problem 9
  ...AD=BC</math> and <math>AB<CD.</math> Suppose that the distances from <math>A</math> to the lines <math>BC,CD,</math> and <math>BD</math> are <math>15,18 pair A, B, C, D, E, F, G, H;
  23 KB (3,640 words) - 18:16, 25 January 2024
• 2021 AIME I Problems/Problem 13
  ...pectively, intersect at distinct points <math>A</math> and <math>B</math>. A third circle <math>\omega</math> is externally tangent to both <math>\omega Let <math>D\equiv\omega\cap\omega_{1}</math> and <math>E\equiv\omega\cap\omega_{2}</math>. The solution relies on the following key
  17 KB (2,852 words) - 03:59, 7 February 2024
• 2023 AIME II Problems
  {{AIME Problems|year=2023|n=II}} ...nd backward. Find the greatest integer less than <math>1000</math> that is a palindrome both when written in base ten and when written in base eight, su
  8 KB (1,370 words) - 21:34, 28 January 2024
• 2022 AIME I Problems/Problem 5
  .... At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their startin ...t <math>B</math> that is <math>14m</math> meters upstream from point <math>A</math>. Similarly, Sherry is swimming downstream for <math>m</math> minutes
  10 KB (1,549 words) - 14:35, 23 February 2023
• 2022 AIME I Problems/Problem 8
  ...where <math>a</math> and <math>b</math> are positive integers. Find <math>a+b.</math> pair A, B, C, W, WA, WB, WC, X, Y, Z;
  12 KB (1,955 words) - 21:11, 31 January 2024
• 2022 AIME I Problems/Problem 11
  ...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> pair A,B,C,D,P,Q;
  16 KB (2,517 words) - 20:22, 31 January 2024
• 2022 AIME I Problems/Problem 14
  ...tting lines of <math>\triangle ABC</math> through <math>M</math> and <math>N</math> intersect at <math>30^\circ.</math> Find the perimeter of <math>\tri ...mple angle chasing reveals the condition is now equivalent to <math>\angle A=120^\circ</math>.
  16 KB (2,730 words) - 02:56, 4 January 2023
• 2021 Fall AMC 10B Problems/Problem 25
  ...and <math>n</math> are relatively prime positive integers. What is <math>m+n?</math> label("$3$",(9/2,3/2),N);
  16 KB (2,274 words) - 09:02, 10 December 2022
• 2021 Fall AMC 12B Problems/Problem 20
  A cube is constructed from <math>4</math> white unit cubes and <math>4</math> <math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\
  10 KB (1,641 words) - 22:17, 26 September 2023
• 2004 IMO Problems/Problem 5
  In a convex quadrilateral <math>ABCD</math>, the diagonal <math>BD</math> bisect Prove that <math>ABCD</math> is a cyclic quadrilateral if and only if <math>AP = CP.</math>
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• 2020 AMC 8 Problems/Problem 21
  ...p paths are there from <math>P</math> to <math>Q?</math> (The figure shows a sample path.) ...i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } for (int i = 0; i < N; ++i) { draw(circle((x[i],y[i])+(0.5,0.5),0.35)); } label("$P$", (5.5, 0.5)
  7 KB (1,170 words) - 22:50, 19 January 2024
• 1982 AHSME Problems/Problem 22
  In a narrow alley of width <math>w</math> a ladder of length <math>a</math> is placed with its foot at point <math>P</math> between the walls. ...wall at <math>Q</math>, the distance <math>k</math> above the ground makes a <math>45^\circ</math> angle with the ground.
  2 KB (383 words) - 22:47, 13 September 2021
• 1977 IMO Problems/Problem 1
  ...egments <math>AK, BK, BL, CL, CM, DM, DN, AN</math> are the 12 vertices of a regular dodecagon. Just use complex numbers, with <math> a = 1</math>, <math> b = i</math>, <math> c = - 1</math> and <math> d = - i</
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• 2021 AMC 12B Problems/Problem 8
  Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</ ...{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math>
  6 KB (878 words) - 00:15, 19 July 2023
• 2021 AMC 12B Problems/Problem 12
  Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> ...(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math>
  5 KB (878 words) - 09:56, 20 May 2023
• 2021 AIME II Problems/Problem 10
  ...and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. triple O1, O2, O3, T1, T2, T3, A, L1, L2;
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• G285 2021 MC10A
  ...ad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2</math> ...the probability that the number selected is a perfect square, given <math>n=4</math>?
  8 KB (1,385 words) - 12:55, 23 June 2021
• G285 2021 Summer Problem Set
  ...\textbf{(B)}\ 43\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 45\qquad\textbf{(E)}\ 46</math> \end{cases}</cmath> If <math>y</math> is a positive integer, find the sum of all values of <math>x</math> such that <m
  9 KB (1,577 words) - 23:28, 28 June 2021
• 2021 Fall AMC 12B Problems/Problem 3
  ...math>2</math> degrees. What is the product of all possible values of <math>N?</math> ...{(B)} \: 30\qquad\textbf{(C)} \: 60\qquad\textbf{(D)} \: 100\qquad\textbf{(E)} \: 120</math>
  2 KB (353 words) - 13:51, 30 December 2022
• 2021 Fall AMC 12B Problems/Problem 23
  What is the average number of pairs of consecutive integers in a randomly selected subset of <math>5</math> distinct integers chosen from th <math>\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{
  15 KB (2,241 words) - 00:38, 6 January 2023
• 2021 Fall AMC 10B Problems/Problem 17
  ...y=0\qquad(\textbf{C}) \: x-3y=0\qquad(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0</math> Even though the problem is phrased as a coordinate bash, that looks disgusting. Instead, let's try to phrase this p
  8 KB (1,331 words) - 22:44, 16 December 2023
• 2021 WSMO Accuracy Round Problems/Problem 5
  ...and <math>c</math> is not divisible by the square of any prime, find <math>a+b+c+d.</math> label("$B$",dir(67.5),N);
  2 KB (369 words) - 10:55, 11 July 2022
• 2021 Fall AMC 12B Problems
  ...\qquad\textbf{(C)} \: 10{,}110\qquad\textbf{(D)} \: 11{,}000\qquad\textbf{(E)} \: 11{,}110</math> ...tbf{(B)} \: 6\qquad\textbf{(C)} \: 8\qquad\textbf{(D)} \: 10\qquad\textbf{(E)} \: 12</math>
  14 KB (2,191 words) - 19:57, 12 November 2023
• 2021 WSMO Team Round Problems
  ...rs? (A word does not have to be an English word, and the word can only use a letter in "Winter Solstice" as many times as it appears) ...and 16 pounds of hay (and still have money left over for tax!). Find <math>n.</math>
  7 KB (1,137 words) - 12:40, 6 June 2022
• 2021 WSMO Speed Round Problems/Problem 5
  ...sitive integers. Find <math>a+b.</math> (A string of letters is defined as a group of consecutive letters with no spaces between them.) ...1 d, 4 e's, 1 l, 2 i's, 1 c, 1 o, 1 u, 1 g, 1 r, 2 s's, 2 n's, 1 b, and 1 a. Thus, there are <math>\frac{19!}{1!4!1!2!1!1!1!1!@!2!1!1!}=\frac{19!}{4!2!
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• 2007 Polish Mathematical Olympiad Third Round
  ...</math> passes the lines <math>AC, BC</math> accordingly in points <math>K,L</math>. Prove that <cmath>\frac{LM}{MK}=\frac{AD}{DB}</cmath>. We call a positive integer <math>\textit{white}</math> if it equals 1 or is the produ
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• AoPSWiki:Sandbox
  A focused asymptote reference with examples and short code.<p style="margin-l * If your code is not working check to make sure you have a semicolon after each line, no extra parentheses or brackets, and no open pa
  28 KB (4,808 words) - 19:43, 3 June 2022
• 2022 AMC 12A Problems/Problem 23
  ...ers <math>1, 2, 3, \ldots, n</math>. For how many integers with <math>1\le{n}\le{22}</math> is <math>k_n<L_n</math>? ...quad\textbf{(B) }3 \qquad\textbf{(C) }7 \qquad\textbf{(D) }8\qquad\textbf{(E) }10</math>
  5 KB (770 words) - 08:52, 29 March 2024
• 2022 AMC 10A Problems/Problem 10
  Daniel finds a rectangular index card and measures its diagonal to be <math>8</math> centi ...2} \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 12\sqrt{2} \qquad \textbf{(E) } 18</math>
  4 KB (553 words) - 12:15, 25 December 2023
• 2022 AMC 12B Problems/Problem 19
  ...positive integer not divisible by the square of any prime. What is <math>m+n+p?</math> ...xtbf{(B) }48 \qquad \textbf{(C) }52 \qquad \textbf{(D) }56 \qquad \textbf{(E) }60</math>
  5 KB (710 words) - 03:53, 12 November 2023
• Complete Quadrilateral
  Let four lines made four triangles of a complete quadrilateral. In the diagram these are <math>\triangle ABC, \tria Let four lines made four triangles of a complete quadrilateral. In the diagram these are <math>\triangle ABC, \tria
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• 2012 EGMO Problems
  ...</math> be a triangle with circumcentre <math>O</math>. The points <math>D,E,F</math> lie in the interiors of the sides <math>BC,CA,AB</math> respective ...h> rows and <math>n</math> columns can be filled with real numbers in such a manner that for any two different rows <math>\left[ {{a_1},{a_2},\ldots,{a_
  5 KB (849 words) - 10:15, 23 December 2022
• Gossard perspector
  ...at the Euler line of a given triangle together with two of its sides forms a triangle whose Euler line is in parallel with the third side of the given t ...d by the Euler line and the sides, taken by two, of a given triangle, form a triangle which is perspective with the given triangle and has the same Eule
  18 KB (3,046 words) - 06:44, 19 January 2023
• 2024 AMC 8 Problems
  ...uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math> ...f{(B) } 6.504\qquad\textbf{(C) } 6.54\qquad\textbf{(D) } 6.9\qquad\textbf{(E) } 6.94</math>
  21 KB (2,856 words) - 17:51, 8 May 2024
• 2016 Mathcounts State Sprint Problems
  ...a+b}</math>. What is the value of <math>5@3</math>? Express your answer as a common fraction. A bag contains <math>4</math> blue, <math>5</math> green and <math>3</math> r
  11 KB (1,672 words) - 21:18, 12 March 2023
• 2023 AMC 10A Problems/Problem 24
  ...below. The distance from any corner of the frame to the nearest vertex of a block is <math>\frac{3}{7}</math> unit. What is the area of the region insi ...rac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}</math>
  13 KB (1,886 words) - 22:08, 10 April 2024
• 2024 AIME I Problems/Problem 14
  ...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n+p</math>. ...(B~(4,5,0)\), \(C~(0,5,8)\), and \(D~(4,0,8)\). Then the plane \(BCD\) has a normal
  13 KB (2,042 words) - 09:34, 3 April 2024
• 2006 iTest Problems/Problem 12
  ...qquad \text{(J) }\frac{2}{12!}\qquad \text{(K) }\frac{1}{20!}\qquad \text{(L) }\text{none of the above}\qquad</math> ...>n\text{th}</math> question has <math>n</math> answer choices, where <math>n</math> goes from <math>1</math> to <math>20</math>)
  2 KB (259 words) - 23:43, 3 November 2023
• 2023 AMC 12A Problems/Problem 13
  In a table tennis tournament every participant played every other participant ex ...uad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66</math>
  5 KB (810 words) - 22:56, 10 May 2024
• 2023 AMC 12B Problems/Problem 25
  ...tices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? ...xtbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}</math>
  19 KB (2,967 words) - 16:56, 24 February 2024
• 2023 SSMO Team Round Problems
  ...2, 3)</math>. Find the largest possible value of <math>a^b + b^c + c^d + d^a</math> A plane and a car start both move northward. The car moves northbound at 60 miles per hou
  8 KB (1,238 words) - 02:14, 3 January 2024
• 2021 WSMO Team Round Problems/Problem 6
  ...ere <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. label("$B$",dir(75),N);
  812 bytes (125 words) - 22:16, 15 December 2023
• DVI exam
  ...State University named after M.V. Lomonosov. The first four problems have a standard level. <cmath>\left\{\begin{array}{l} 2 x^2 + 4xy + 11 y^2 \le 1,\\4x + 7y \ge 3.\end{array}\right.</cmath>
  32 KB (5,375 words) - 11:43, 5 May 2024
• Problems Collection
  This is a page where you can share the problems you made (try not to use past exams). ...h> are positive integers. If <math>m^2=2^8+2^{11}+2^n</math>, find <math>m+n</math>.
  43 KB (7,006 words) - 14:24, 19 February 2024
• 2024 AIME II Problems/Problem 10
  ...vely. Furthermore, extend <math>AI</math> to meet <math>BC</math> at <math>L</math> and the circumcircle of triangle <math>ABC</math> at <math>D</math>. real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
  17 KB (2,854 words) - 15:18, 23 April 2024
• 2024 AIME II Problems/Problem 12
  ...-axis. There is a unique point \(C\) on \(\overline{AB}\), distinct from \(A\) and \(B\), that does not belong to any segment from \(\mathcal{F}\) othe ...critical condition that must be satisfied by our parameters is that <math>a^2+b^2=1</math>, since the length of <math>\overline{PQ}=1</math>.
  13 KB (2,178 words) - 11:34, 14 May 2024
• 2005 iTest Problems
  Carrie, Miranda, Charlotte, and Samantha are sitting at a table with <math>5</math> numbered chairs (numbered <math>1</math> through ...her teacher just couldn’t contain her applause, and awarded an automatic A to her for the rest of the year.
  25 KB (3,738 words) - 10:53, 23 April 2024
• Power of a point theorem
  ...e point. Can be useful with [[cyclic quadrilaterals]] as well however with a slightly different application. ...</math> and <math> CD </math> intersect at a point <math> P </math> within a circle, then <math> AP\cdot BP=CP\cdot DP </math>
  6 KB (1,010 words) - 02:38, 7 May 2024